THE PROBLEM OF INFINITY 163 



Thus the length of the side and the length of the 

 diagonal are incommensurable ; that is to say, however 

 small a unit of length you take, if it is contained an exact 

 number of times in the side, it is not contained any exact 

 number of times in the diagonal, and vice versa. 



Now this fact might have been assimilated by some 

 philosophies without any great difficulty, but to the 

 philosophy of Pythagoras it was absolutely fatal. Pytha- 

 goras held that number is the constitutive essence of 

 all things, yet no two numbers could express the ratio 

 of the side of a square to the diagonal. It would seem 

 probable that we may expand his difficulty, without 

 departing from his thought, by assuming that he regarded 

 the length of a line as determined by the number of atoms 

 contained in it a line two inches long would contain 

 twice as many atoms as a line one inch long, and so on. 

 But if this were the truth, then there must be a definite 

 numerical ratio between any two finite lengths, because 

 it was supposed that the number of atoms in each, how- 

 ever large, must be finite. Here there was an insoluble 

 contradiction. The Pythagoreans, it is said, resolved to 

 keep the existence of incommensurables a profound 

 secret, revealed only to a few of the supreme heads 

 of the sect ; and one of their number, Hippasos of 

 Metapontion, is even said to have been shipwrecked at 

 sea for impiously disclosing the terrible discovery to their 

 enemies. It must be remembered that Pythagoras was the 

 founder of a new religion as well as the teacher of a new 

 science : if the science came to be doubted, the disciples 

 might fall into sin, and perhaps even eat beans, which 

 according to Pythagoras is as bad as eating parents' bones. 



be even. But, since m is even, and m and n have no common factor, n 

 must be odd. Thus n must be both odd and even, which is impossible ; 

 and therefore the diagonal and the side cannot have a rational ratio. 



